This article first discusses the finite-dimensional case and its applications before considering compact operators on infinite-dimensional Hilbert spaces. We will see that for compact operators, the proof of the main theorem uses essentially the same idea from the finite-dimensional argument.

In the case that the operator is non-Hermitian, the theorem provides an equivalent characterization of the associated singular values. The min-max theorem can be extended to self-adjoint operators that are bounded below.

where (⋅, ⋅) denotes the Euclidean inner product on Cn. Clearly, the Rayleigh quotient of an eigenvector is its associated eigenvalue. Equivalently, the Rayleigh–Ritz quotient can be replaced by

f(x)=(Ax,x),‖x‖=1.{\displaystyle f(x)=(Ax,x),\;\|x\|=1.}

For Hermitian matrices, the range of the continuous function RA(x), or f(x), is a compact subset [a, b] of the real line. The maximum b and the minimum a are the largest and smallest eigenvalue of A, respectively. The min-max theorem is a refinement of this fact.

Since the matrix A is Hermitian it is diagonalizable and we can choose an orthonormal basis of eigenvectors {u1, ..., un} that is, ui is an eigenvector for the eigenvalue λi and such that (ui, ui) = 1 and (ui, uj) = 0 for all i ≠ j.

If U is a subspace of dimension k then its intersection with the subspace span{uk, ..., un} isn't zero (by simply checking dimensions) and hence there exists a vector v ≠ 0 in this intersection that we can write as

In the case where U is a subspace of dimension n-k+1, we proceed in a similar fashion: Consider the subspace of dimension k, span{u1, ..., uk}. Its intersection with the subspace U isn't zero (by simply checking dimensions) and hence there exists a vector v in this intersection that we can write as

Again, this is one part of the equation. To get the other inequality, note again that the eigenvector u of
λk{\displaystyle \lambda _{k}} is contained in U = span{uk, ..., un} so that we can conclude the equality.

Define the Rayleigh quotient RN(x){\displaystyle R_{N}(x)} exactly as above in the Hermitian case. Then it is easy to see that the only eigenvalue of N is zero, while the maximum value of the Rayleigh ratio is 1/2. That is, the maximum value of the Rayleigh quotient is larger than the maximum eigenvalue.

The singular values {σk} of a square matrix M are the square roots of the eigenvalues of M*M (equivalently MM*). An immediate consequence[citation needed] of the first equality in the min-max theorem is:

Let A be a symmetric n × n matrix. The m × m matrix B, where m ≤ n, is called a compression of A if there exists an orthogonal projectionP onto a subspace of dimension m such that P*AP = B. The Cauchy interlacing theorem states:

Theorem. If the eigenvalues of A are α1 ≤ ... ≤ αn, and those of B are β1 ≤ ... ≤ βj ≤ ... ≤ βm, then for all j ≤ m,

Let A be a compact, Hermitian operator on a Hilbert space H. Recall that the spectrum of such an operator (the set of eigenvalues) is a set of real numbers whose only possible cluster point is zero. It is thus convenient to list the positive eigenvalues of A as

where entries are repeated with multiplicity, as in the matrix case. (To emphasize that the sequence is decreasing, we may write λk=λk↓{\displaystyle \lambda _{k}=\lambda _{k}^{\downarrow }}.) When H is infinite-dimensional, the above sequence of eigenvalues is necessarily infinite. We now apply the same reasoning as in the matrix case. Letting Sk ⊂ H be a k dimensional subspace, we can obtain the following theorem.

(Click "show" at right to see the proof of this theorem or "hide" to hide it.)

Let S' be the closure of the linear span S′=span⁡{uk,uk+1,…}{\displaystyle S'=\operatorname {span} \{u_{k},u_{k+1},\ldots \}}.
The subspace S' has codimension k − 1. By the same dimension count argument as in the matrix case, S' ∩ Sk is non empty. So there exists x ∈ S' ∩ Sk with ‖x‖=1{\displaystyle \|x\|=1}. Since it is an element of S' , such an x necessarily satisfy

The min-max theorem also applies to (possibly unbounded) self-adjoint operators.[1][2] Recall the essential spectrum is the spectrum without isolated eigenvalues of finite multiplicity. Sometimes we have some eigenvalues below the essential spectrum, and we would like to approximate the eigenvalues and eigenfunctions.

Theorem (Min-Max). Let A be self-adjoint, and let E1≤E2≤E3≤⋯{\displaystyle E_{1}\leq E_{2}\leq E_{3}\leq \cdots } be the eigenvalues of A below the essential spectrum. Then

If we only have N eigenvalues and hence run out of eigenvalues, then we let En:=infσess(A){\displaystyle E_{n}:=\inf \sigma _{ess}(A)} (the bottom of the essential spectrum) for n>N, and the above statement holds after replacing min-max with inf-sup.

Theorem (Max-Min). Let A be self-adjoint, and let E1≤E2≤E3≤⋯{\displaystyle E_{1}\leq E_{2}\leq E_{3}\leq \cdots } be the eigenvalues of A below the essential spectrum. Then

If we only have N eigenvalues and hence run out of eigenvalues, then we let En:=infσess(A){\displaystyle E_{n}:=\inf \sigma _{ess}(A)} (the bottom of the essential spectrum) for n>N, and the above statement holds after replacing max-min with sup-inf.

The proofs[1][2] use the following results about self-adjoint operators:

Theorem. Let A be self-adjoint. Then (A−E)≥0{\displaystyle (A-E)\geq 0} for E∈R{\displaystyle E\in \mathbb {R} } if and only if σ(A)⊆[E,∞){\displaystyle \sigma (A)\subseteq [E,\infty )}.